Were there any direct or immediate financial/banking policies made because of this recession? I've read that a combination of factors helped get America out of the Panic, but it never mentioned a reform in those policies.
The Panic of 1837 proved that the "cure was worse than the disease." Thus, it set the U.S. on the road to the passage of the 16th Amendment, and the establishment of the Federal Reserve some 76 years later, in 1913.
The "disease" that Jackson fought was a centralized national bank along the lines of the Bank of England, advocated by banker Nicholas Biddle.
The Jacksonian "cure" for America's banking woes was a number of small, decentralized, and undercapitalized banks, many of which collapsed in the Panic of 1837, leading to a depression. This experience convinced many Americans that they needed a central, or at least a very large, bank.
Around the turn of the 19th and 20th centuries J.P.Morgan created a large bank that bore his name (until the merger with Chase) that the Europeans considered the "unofficial" central bank of the United States. But the Crash of 1907 convinced the country that it needed an even stronger central bank, leading to the 16th Amendment to the Constitution providing for a central currency, the Federal Reserve, and the income tax.
Specie Circular: The most direct and immediate effect of the Panic of 1837 was the repeal of the Specie Circular of 1836. The Specie Circular was a highly deflationary policy because it required that Western lands be purchased with specie that just wasn't available. Deflation exacerbates recessions: never good. In May 1838, the Senate repealed the Specie Circular by a vote 34 to 9, and the House repealed it by a vote of 154 to 29.
Independent Treasury: However, Democrats were firmly back in control of the national government by 1846. They then created the Independent Treasury System, which was an attempt to separate government finances from what they saw as the corrupting influence of the national banking system (and so was a complete reversal from the pet bank system).
The Independent Treasury also had a largely deflationary effect on the American economy, and so should have caused troubles for the Democrats. However, due to complete coincidence, these effects were more than counteracted by the inflationary effects of the Gold Rush of 1848, which flooded the U.S. with hard currency.
In short, Americans learned few valuable lessons from the Panic of 1837 because observers couldn't properly discern the relationship between causes and effects: financial policies were too quick to come and go through the 1830s and 1840s, and there were too many confounding factors like the Gold Rush. That's why (as Tom Au notes), the US had to experience several more Panics and Crashes before settling on the Federal Reserve System some 76 years later.
Arguably the most important long-term effect of the Panic of 1837: States had been borrowing like mad to fund canals, railroads, and banks as "mixed" public-private corporations. The Panic, being deflationary, increased the real value of the states' debts at the same time as it decreased their tax revenues. This led to a full-fledged bankruptcy crisis. States washed their hands of the "mixed" corporation, and from this time forward corporations began to take on an increasingly "private" character. So the Panic of 1837 is an under-appreciated factor in explaining why American political economy looks so different from European political economy. (Some great books on this claim by Louis Hartz and William Roy.)